For After Effects, you want a lot of RAM, several processors (CPUs), and at least two fast local disk drives.
Since you referred to the suite, I'll gues that you're using Premiere Pro CS5, too. If so, you'll want one of the NVIDIA graphics cards that supports CUDA acceleration features of Premiere Pro CS5. For After Effects, the graphics card isn't really relevant.
So, when I say "a lot of RAM", how much do I mean? The more, the better.
I am very happy with my quad-core computer (eight CPUs, when hyperthreading is taken into account) with 24GB of RAM and a Quadro CX card. This is an entirely adequate system for editing RED footage with Premiere Pro CS5 and doing large-frame compositing work with After Effects. Yes, you can make do with less, but I'm just trying to give an idea of what works great with no hiccups for me.
Some insight into hardware usage for After Effects can be gathered here:
And for Premiere Pro, you should really ask on the Premeire Pro Hardware Forum: